Stark's conjecture
Encyclopedia
In number theory
, the Stark conjectures, the first of which were introduced in a series of papers in the 1970s by American
mathematician Harold Stark
and later greatly expanded by John Tate
, give conjectural
information about the coefficient
of the leading term in the Taylor expansion of an Artin L-function
associated with a Galois extension
K/k of algebraic number field
s. The conjectures give a vast generalization of the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function
of a number field as the product of a regulator related to S-units of the field and a rational number
. When K/k is an abelian extension
and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units. Extensions of this refined conjecture to higher orders of vanishing have been provided by Karl Rubin
and Cristian Dumitru Popescu.
. When the extension is abelian
and the order of vanishing of an L-function at s = 0 is one, Stark's refined conjecture predicts the existence of the Stark units, whose roots generate Kummer extensions of K that are abelian over the base field k (and not just abelian over K, as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving Hilbert's twelfth problem
. Also, it is possible to compute Stark units in specific examples, allowing verification of the veracity of his refined conjecture as well as providing an important computational tool for generating abelian extensions of number fields. In fact, some standard algorithms for computing abelian extensions of number fields involve producing Stark units that generate the extensions (see below)
s of totally real number fields, and the conjectures provide one solution to Hilbert's twelfth problem, which challenged mathematicians to show how class fields
may be constructed over any number field by the methods of complex analysis
.
, the abelian Stark conjectures are still unproved in number fields, and more progress has been made in function fields of an algebraic variety
.
The works of Yuri I. Manin
related Stark's conjectures to noncommutative geometry
of Alain Connes
. This provides a very attractive conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Stark conjectures, the first of which were introduced in a series of papers in the 1970s by American
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician Harold Stark
Harold Stark
Harold Mead Stark is an American mathematician, specializing in number theory. He is best known for his solution of the Gauss class number 1 problem, in effect correcting and completing the earlier work of Kurt Heegner; and for Stark's conjecture. He has recently collaborated with Audrey Terras on...
and later greatly expanded by John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
, give conjectural
Conjecture
A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...
information about the coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
of the leading term in the Taylor expansion of an Artin L-function
Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin...
associated with a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
K/k of algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
s. The conjectures give a vast generalization of the analytic class number formula expressing the leading coefficient of the Taylor series for the Dedekind zeta function
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...
of a number field as the product of a regulator related to S-units of the field and a rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
. When K/k is an abelian extension
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
and the order of vanishing of the L-function at s = 0 is one, Stark gave a refinement of his conjecture, predicting the existence of certain S-units, called Stark units. Extensions of this refined conjecture to higher orders of vanishing have been provided by Karl Rubin
Karl Rubin
Karl Rubin is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. His research interest is in elliptic curves. He was the first mathematician to show that some elliptic curves over the rationals have finite Tate-Shafarevich groups...
and Cristian Dumitru Popescu.
Formulation
The Stark conjectures, in the most general form, predict that the leading coefficient of an Artin L-function is the product of a type of regulator, the Stark regulator, with an algebraic numberAlgebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
. When the extension is abelian
Abelian extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
and the order of vanishing of an L-function at s = 0 is one, Stark's refined conjecture predicts the existence of the Stark units, whose roots generate Kummer extensions of K that are abelian over the base field k (and not just abelian over K, as Kummer theory implies). As such, this refinement of his conjecture has theoretical implications for solving Hilbert's twelfth problem
Hilbert's twelfth problem
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field...
. Also, it is possible to compute Stark units in specific examples, allowing verification of the veracity of his refined conjecture as well as providing an important computational tool for generating abelian extensions of number fields. In fact, some standard algorithms for computing abelian extensions of number fields involve producing Stark units that generate the extensions (see below)
Computation
The first order zero conjectures are used in recent versions of the PARI/GP computer algebra system to compute Hilbert class fieldHilbert class field
In algebraic number theory, the Hilbert class field E of a number field K is the maximal abelian unramified extension of K. Its degree over K equals the class number of K and the Galois group of E over K is canonically isomorphic to the ideal class group of K using Frobenius elements for prime...
s of totally real number fields, and the conjectures provide one solution to Hilbert's twelfth problem, which challenged mathematicians to show how class fields
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
may be constructed over any number field by the methods of complex analysis
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
.
Progress
Stark's principal conjecture has been proven in various special cases, including the case where the character defining the L-function takes on only rational values. Except when the base field is the field of rational numbers or an imaginary quadratic fieldQuadratic field
In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q. It is easy to show that the map d ↦ Q is a bijection from the set of all square-free integers d ≠ 0, 1 to the set of all quadratic fields...
, the abelian Stark conjectures are still unproved in number fields, and more progress has been made in function fields of an algebraic variety
Function field of an algebraic variety
In algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
.
The works of Yuri I. Manin
Yuri I. Manin
Yuri Ivanovitch Manin is a Soviet/Russian/German mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.-Biography:...
related Stark's conjectures to noncommutative geometry
Noncommutative geometry
Noncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
of Alain Connes
Alain Connes
Alain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.-Work:...
. This provides a very attractive conceptual framework for studying the conjectures, although at the moment it is unclear whether Manin's techniques will yield the actual proof.